Condition i. implies that \(Q_n\)  is strongly convex with probability 1 while condition ii. requires Lipschitz continuity of the Hessian.A function \(Q_n\)  is strongly convex on \(\Theta \)  if for all \(\theta \in \Theta \) , there exists some \(\underline{\lambda }>0\)  such that \(\nabla ^2 Q_n(\theta ) \ge \underline{\lambda } I_d\) . For bounded \(\Theta \) , there also exists \(\overline{\lambda }\)  such that \(\nabla ^2 Q_n(\theta ) \le \overline{\lambda } I_d\) . Then \(\overline{\lambda }/\underline{\lambda }\)  is an upper bound on the condition number of \(\nabla ^2 Q_n(\theta )\) . Strong convexity of the sample objective function can be less restrictive than it appears as some non-convex problems can be convexified by penalization.In some cases, the strict convexity assumption restricts the parameter space to be bounded in the absence of penalization. See e.g. the discussion of Assumption (H3) in [1]}. Assumption REF  implies the so-called Polyak-Łojasiewicz inequalities:
\( \langle \theta - \hat{\theta }_n, G_n(\theta ) \rangle &= (\theta - \hat{\theta }_n)^\prime H_n(\tilde{\theta }_n) (\theta - \hat{\theta }_n) \ge \underline{\lambda }_{H} \Vert \theta -\hat{\theta }_n\Vert _2^2, \\\Vert  G_n(\theta ) \Vert _2^2 &= (\theta - \hat{\theta }_n)^\prime H_n(\tilde{\theta }_n)^2 (\theta - \hat{\theta }_n) \le \overline{\lambda }_H^2 \Vert \theta -\hat{\theta }_n\Vert ^2_2,\) 
